Thursday, April 26, 2018

Alethic Platonism

I’ve been thinking about an interesting metaphysical thesis about arithmetic, which we might call alethic Platonism about arithmetic: there is a privileged, complete and objectively correct assignment of truth values to arithmetical sentences, not relative to a particular model or axiomatization.

Prima facie, one can be an alethic Platonist about arithmetic without being an ontological Platonist: one can be an alethic Platonist without thinking that numbers really exist. One might, for instance, be a conceptualist, or think that facts about natural numbers are hypothetical facts about sequences of dashes.

Conversely, one can be an ontological Platonist without being an alethic Platonist about arithmetic: one can, for instance, think there really are infinitely many pluralities of abstracta each of which is equally well qualified to count as “the natural numbers”, with different such candidates for “the natural numbers” disagreeing on some of the truths of arithmetic.

Alethic Platonism is, thus, orthogonal to ontological Platonism. Similar orthogonal pairs of Platonist claims can be made about sets as about naturals.

One might also call alethic Platonism “alethic absolutism”.

I suspect causal finitism commits one to alethic Platonism.

Something close to alethic Platonism about arithmetic is required if one thinks that there is a privileged, complete and objectively correct assignment of truth values to claims about what sentence can be proved from what sentence. Specifically, it seems to me that such an absolutism about proof-existence commits one to alethic Platonism about the Σ10 sentences of arithmetic.


Philip Rand said...

Cantor beat you to it...

Alexander R Pruss said...

I expect Cantor didn't know that there was no complete consistent recursive axiomatization of arithmetic. Prior to that piece of knowledge, it is difficult to distinguish alethic Platonism from other views of mathematics, like certain versions of formalism.

Philip Rand said...

No, Cantor was quite clever... after all it was he who came up with the idea of the height of an equation and from there schemes... from there one can relate grammar to the polynomial which leads inevitably to ontological Platonism (if one is so inclined).

Philip Rand said...

Interestingly, related to hierarchy of equations; such a phenomenon is seen in the origin of the Lagrangian we use in physics and what the squared gradient term in all Lagrangians represents.